Field theory is the branch of
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
in which
fields are studied. This is a glossary of some terms of the subject. (See
field theory (physics) for the unrelated field theories in physics.)
Definition of a field
A field is a
commutative ring
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not sp ...
(''F'',+,*) in which 0≠1 and every nonzero element has a multiplicative inverse. In a field we thus can perform the operations addition, subtraction, multiplication, and division.
The non-zero elements of a field ''F'' form an
abelian group under multiplication; this group is typically denoted by ''F''
×;
The
ring of polynomials in the variable ''x'' with coefficients in ''F'' is denoted by ''F''
'x''
Basic definitions
;
Characteristic : The ''characteristic'' of the field ''F'' is the smallest positive
integer ''n'' such that ''n''·1 = 0; here ''n''·1 stands for ''n'' summands 1 + 1 + 1 + ... + 1. If no such ''n'' exists, we say the characteristic is zero. Every non-zero characteristic is a
prime number. For example, the
rational numbers, the
real numbers and the
''p''-adic numbers have characteristic 0, while the finite field Z
''p'' where ''p'' is prime has characteristic ''p''.
; Subfield : A ''subfield'' of a field ''F'' is a
subset
In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
of ''F'' which is closed under the field operation + and * of ''F'' and which, with these operations, forms itself a field.
;
Prime field : The ''prime field'' of the field ''F'' is the unique smallest subfield of ''F''.
;
Extension field : If ''F'' is a subfield of ''E'' then ''E'' is an ''extension field'' of ''F''. We then also say that ''E''/''F'' is a ''field extension''.
;
Degree of an extension : Given an extension ''E''/''F'', the field ''E'' can be considered as a
vector space over the field ''F'', and the
dimension of this vector space is the ''degree'' of the extension, denoted by
'E'' : ''F''
; Finite extension : A ''finite extension'' is a field extension whose degree is finite.
;
Algebraic extension : If an element α of an extension field ''E'' over ''F'' is the
root of a non-zero polynomial in ''F''
'x'' then α is ''algebraic'' over ''F''. If every element of ''E'' is algebraic over ''F'', then ''E''/''F'' is an ''algebraic extension''.
; Generating set : Given a field extension ''E''/''F'' and a subset ''S'' of ''E'', we write ''F''(''S'') for the smallest subfield of ''E'' that contains both ''F'' and ''S''. It consists of all the elements of ''E'' that can be obtained by repeatedly using the operations +,−,*,/ on the elements of ''F'' and ''S''. If ''E'' = ''F''(''S'') we say that ''E'' is generated by ''S'' over ''F''.
;
Primitive element : An element α of an extension field ''E'' over a field ''F'' is called a ''primitive element'' if ''E''=''F''(α), the smallest extension field containing α. Such an extension is called a
simple extension.
;
Splitting field : A field extension generated by the complete factorisation of a polynomial.
;
Normal extension : A field extension generated by the complete factorisation of a set of polynomials.
;
Separable extension : An extension generated by roots of
separable polynomials.
;
Perfect field : A field such that every finite extension is separable. All fields of characteristic zero, and all finite fields, are perfect.
;
Imperfect degree : Let ''F'' be a field of characteristic ''p''>0; then ''F''
''p'' is a subfield. The degree
''p''">'F'':''F''''p''is called the ''imperfect degree'' of ''F''. The field ''F'' is perfect if and only if its imperfect degree is ''1''. For example, if ''F'' is a function field of ''n'' variables over a finite field of characteristic ''p''>0, then its imperfect degree is ''p''
n.
[Fried & Jarden (2008) p.45]
;
Algebraically closed field
In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in .
Examples
As an example, the field of real numbers is not algebraically closed, because ...
: A field ''F'' is ''algebraically closed'' if every polynomial in ''F''
'x''has a root in ''F''; equivalently: every polynomial in ''F''
'x''is a product of linear factors.
;
Algebraic closure: An ''algebraic closure'' of a field ''F'' is an algebraic extension of ''F'' which is algebraically closed. Every field has an algebraic closure, and it is unique up to an isomorphism that fixes ''F''.
;
Transcendental
Transcendence, transcendent, or transcendental may refer to:
Mathematics
* Transcendental number, a number that is not the root of any polynomial with rational coefficients
* Algebraic element or transcendental element, an element of a field exten ...
: Those elements of an extension field of ''F'' that are not algebraic over ''F'' are ''transcendental'' over ''F''.
; Algebraically independent elements : Elements of an extension field of ''F'' are ''algebraically independent'' over ''F'' if they don't satisfy any non-zero polynomial equation with coefficients in ''F''.
;
Transcendence degree : The number of algebraically independent transcendental elements in a field extension. It is used to define the
dimension of an algebraic variety.
Homomorphisms
; Field homomorphism : A ''field homomorphism'' between two fields ''E'' and ''F'' is a
function
::''f'' : ''E'' → ''F''
:such that, for all ''x'', ''y'' in ''E'',
::''f''(''x'' + ''y'') = ''f''(''x'') + ''f''(''y'')
::''f''(''xy'') = ''f''(''x'') ''f''(''y'')
::''f''(1) = 1.
:These properties imply that , for ''x'' in ''E'' with , and that ''f'' is
injective
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositiv ...
. Fields, together with these homomorphisms, form a
category. Two fields ''E'' and ''F'' are called isomorphic if there exists a
bijective homomorphism
::''f'' : ''E'' → ''F''.
:The two fields are then identical for all practical purposes; however, not necessarily in a ''unique'' way. See, for example,
complex conjugation
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
.
Types of fields
;
Finite field : A field with finitely many elements. Aka Galois field.
;
Ordered field : A field with a
total order compatible with its operations.
;
Rational numbers
;
Real numbers
;
Complex numbers
;
Number field : Finite extension of the field of rational numbers.
;
Algebraic number
An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio, (1 + \sqrt)/2, is an algebraic number, because it is a root of the po ...
s : The field of algebraic numbers is the smallest algebraically closed extension of the field of rational numbers. Their detailed properties are studied in
algebraic number theory
Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic ob ...
.
;
Quadratic field : A degree-two extension of the rational numbers.
;
Cyclotomic field : An extension of the rational numbers generated by a
root of unity.
;
Totally real field : A number field generated by a root of a polynomial, having all its roots real numbers.
;
Formally real field
;
Real closed field
;
Global field : A number field or a function field of one variable over a finite field.
;
Local field : A completion of some global field (
w.r.t.
(Main list of acronyms)
__NOTOC__
* W
** (s) Tungsten (from German ''Wolfram'')
** (i) Watt
** (i) Wednesday
** (i) West
** (i) Games Won in sporting competitions
W0–9
* W3C – (i) World Wide Web Consortium
* w/ – (p) With: ''Softwa ...
a prime of the integer ring).
;
Complete field In mathematics, a complete field is a field equipped with a metric and complete with respect to that metric. Basic examples include the real numbers, the complex numbers, and complete valued fields (such as the ''p''-adic numbers).
Constructio ...
: A field complete w.r.t. to some valuation.
;
Pseudo algebraically closed field : A field in which every variety has a
rational point.
[Fried & Jarden (2008) p.214]
;
Henselian field In mathematics, a Henselian ring (or Hensel ring) is a local ring in which Hensel's lemma holds. They were introduced by , who named them after Kurt Hensel. Azumaya originally allowed Henselian rings to be non-commutative, but most authors now rest ...
: A field satisfying
Hensel lemma In mathematics, Hensel's lemma, also known as Hensel's lifting lemma, named after Kurt Hensel, is a result in modular arithmetic, stating that if a univariate polynomial has a simple root modulo a prime number , then this root can be ''lifted'' to a ...
w.r.t. some valuation. A generalization of complete fields.
;
Hilbertian field: A field satisfying
Hilbert's irreducibility theorem: formally, one for which the
projective line is not
thin in the sense of Serre.
[Serre (1992) p.19][Schinzel (2000) p.298]
; Kroneckerian field: A totally real algebraic number field or a totally imaginary quadratic extension of a totally real field.
[Schinzel (2000) p.5]
;
CM-field or J-field: An algebraic number field which is a totally imaginary quadratic extension of a totally real field.
;
Linked field In mathematics, a linked field is a field for which the quadratic forms attached to quaternion algebras have a common property.
Linked quaternion algebras
Let ''F'' be a field of characteristic not equal to 2. Let ''A'' = (''a''1,''a''2) and ''B ...
: A field over which no
biquaternion algebra In mathematics, a biquaternion algebra is a compound of quaternion algebras over a field.
The biquaternions of William Rowan Hamilton (1844) and the related split-biquaternions and dual quaternions do not form biquaternion algebras in this sense.
...
is a
division algebra.
[Lam (2005) p.342]
; Frobenius field: A
pseudo algebraically closed field whose
absolute Galois group has the embedding property.
[Fried & Jarden (2008) p.564]
Field extensions
Let ''E''/''F'' be a field extension.
;
Algebraic extension : An extension in which every element of ''E'' is algebraic over ''F''.
;
Simple extension: An extension which is generated by a single element, called a primitive element, or generating element. The
primitive element theorem classifies such extensions.
;
Normal extension : An extension that splits a family of polynomials: every root of the minimal polynomial of an element of ''E'' over ''F'' is also in ''E''.
;
Separable extension : An algebraic extension in which the minimal polynomial of every element of ''E'' over ''F'' is a
separable polynomial, that is, has distinct roots.
[Fried & Jarden (2008) p.28]
;
Galois extension : A normal, separable field extension.
;
Primary extension : An extension ''E''/''F'' such that the algebraic closure of ''F'' in ''E'' is
purely inseparable In algebra, a purely inseparable extension of fields is an extension ''k'' ⊆ ''K'' of fields of characteristic ''p'' > 0 such that every element of ''K'' is a root of an equation of the form ''x'q'' = ''a'', wit ...
over ''F''; equivalently, ''E'' is
linearly disjoint from the
separable closure of ''F''.
[Fried & Jarden (2008) p.44]
;
Purely transcendental extension
In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ...
: An extension ''E''/''F'' in which every element of ''E'' not in ''F'' is transcendental over ''F''.
[
; ]Regular extension In field theory, a branch of algebra, a field extension L/k is said to be regular if ''k'' is algebraically closed in ''L'' (i.e., k = \hat k where \hat k is the set of elements in ''L'' algebraic over ''k'') and ''L'' is separable over ''k'', or ...
: An extension ''E''/''F'' such that ''E'' is separable over ''F'' and ''F'' is algebraically closed in ''E''.[
; ]Simple radical extension
In mathematics and more specifically in field theory, a radical extension of a field ''K'' is an extension of ''K'' that is obtained by adjoining a sequence of ''n''th roots of elements.
Definition
A simple radical extension is a simple extensi ...
: A simple extension ''E''/''F'' generated by a single element α satisfying for an element ''b'' of ''F''. In characteristic ''p'', we also take an extension by a root of an Artin–Schreier polynomial to be a simple radical extension.[Roman (2007) p.273]
; Radical extension: A tower where each extension is a simple radical extension.[
; Self-regular extension : An extension ''E''/''F'' such that ''E'' ⊗''F'' ''E'' is an integral domain.
; Totally transcendental extension: An extension ''E''/''F'' such that ''F'' is algebraically closed in ''F''.]
; Distinguished class: A class ''C'' of field extensions with the three properties[Lang (2002) p.228]
:# If ''E'' is a C-extension of ''F'' and ''F'' is a C-extension of ''K'' then ''E'' is a C-extension of ''K''.
:# If ''E'' and ''F'' are C-extensions of ''K'' in a common overfield ''M'', then the compositum
In mathematics, the tensor product of two fields is their tensor product as algebras over a common subfield. If no subfield is explicitly specified, the two fields must have the same characteristic and the common subfield is their prime su ...
''EF'' is a C-extension of ''K''.
:# If ''E'' is a C-extension of ''F'' and ''E'' > ''K'' > ''F'' then ''E'' is a C-extension of ''K''.
Galois theory
; Galois extension : A normal, separable field extension.
; Galois group : The automorphism group of a Galois extension. When it is a finite extension, this is a finite group of order equal to the degree of the extension. Galois groups for infinite extensions are profinite group In mathematics, a profinite group is a topological group that is in a certain sense assembled from a system of finite groups.
The idea of using a profinite group is to provide a "uniform", or "synoptic", view of an entire system of finite groups. ...
s.
; Kummer theory : The Galois theory of taking ''n''-th roots, given enough roots of unity. It includes the general theory of quadratic extensions.
; Artin–Schreier theory : Covers an exceptional case of Kummer theory, in characteristic ''p''.
; Normal basis : A basis in the vector space sense of ''L'' over ''K'', on which the Galois group of ''L'' over ''K'' acts transitively.
; Tensor product of fields : A different foundational piece of algebra, including the compositum
In mathematics, the tensor product of two fields is their tensor product as algebras over a common subfield. If no subfield is explicitly specified, the two fields must have the same characteristic and the common subfield is their prime su ...
operation ( join of fields).
Extensions of Galois theory
; Inverse problem of Galois theory : Given a group ''G'', find an extension of the rational number or other field with ''G'' as Galois group.
; Differential Galois theory : The subject in which symmetry groups of differential equations are studied along the lines traditional in Galois theory. This is actually an old idea, and one of the motivations when Sophus Lie founded the theory of Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
s. It has not, probably, reached definitive form.
; Grothendieck's Galois theory : A very abstract approach from algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, introduced to study the analogue of the fundamental group
In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...
.
References
*
*
*
*
*
*
*
*
*
{{DEFAULTSORT:Glossary Of Field Theory
Field theory
*
Wikipedia glossaries using description lists